<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

<html>
<head>
<title>Simulations for Statistical and Thermal Physics</title>

<link href="../../default.css" type="text/css" rel="stylesheet">

</head>

<body>
<h3 style="text-align:center;">Molecular dynamics of hard disks</h3>

<p class="header_title">Introduction</p>

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.harddisk.HDApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p>The algorithm for simulating the evolution of a system of hard disks can
be summarized by the following:</p>

<ol>
<li>For each particle find the particle that it would collide with next and record the collision time.</li>

<li>Determine the minimum collision time.</li>

<li>Move all the particles using a straight line trajectory for a time
equal to the minimum collision time.</li>

<li>Update the collision times.</li>

<li>Compute the post collision velocities of the two particles that collide.</li>

<li>Calculate any quantities of interest and accumulate data.</li>

<li>Update the collision partners for the colliding pair and any other particle 
that would have collided with either of the colliding pair, if they had not collided.</li>

</ol>

<p>Because hard disks
move in straight lines at constant speed between collisions and
change their velocities instantaneously when a collision occurs, the problem becomes finding the next collision and computing
<i>event driven</i> and can be computed exactly in principle; in practice,
it is limited only by roundoff errors.</p>

<p class="header_title">Problems</p>

<ol>

<li> Run the simulation using the default parameters. Does the temperature 
fluctuate during the simulation? What is the effect of doubling the initial 
temperature without changing any of the other parameters? Is the temperature a 
useful quantity for hard disks?</li>

<li>Each point of the plot of pA/NkT is averaged over ten collisions. 
Does the pressure P fluctuate? Compute PA/NkT as a function of the density for a fixed 
number of particles. Is there any evidence
of a phase transition where the slope of the pressure with respect to density changes 
abruptly?</li>

<li>The mean free path and the mean collision time are also computed. Show that the mean free 
path is independent of temperature, and the mean collision time is proportional to the 
square root of temperature. How does the mean free path and collision time change if you double the linear dimensions of the simulation cell?</li>

<li>*We can gain more insight into the structure of a many body system by looking
at how the positions of the particles are correlated with one another due to
their interactions. The
<i>radial distribution function</i> g(r) is a measure of this
correlation. Suppose that
N particles are in a region of volume V with number density &#961; = N/V.
Choose one of the particles to be the origin. Then the mean number of other
particles between <b>r</b> and <b>r</b> + d<b>r</b> is defined to be
&#961;g(<b>r</b>)d<b>r</b>. If the
interparticle interaction is spherically symmetric and the system is
a gas or a liquid, then g(<b>r</b>) depends only on the
separation r = |<b>r</b>|. The normalization condition for g(r) is
<p class="center">
&#961;&#8747;g(r)d<b>r</b> = N - 1 &#8773; N, 
</p>
where the volume element d<b>r</b> = 4&#960;r<sup>2</sup>dr
(d = 3), 2&#960;rdr, (d = 2), and 2dr (d = 1).
To determine g(r) for a particular configuration of particles, we 
compute n(r,&#916;r), the number of particles in a spherical (circular) shell
of radius r and a small, but nonzero width
&#916;r, with the center of the shell centered about each particle. We then properly normalize g(r). What is the qualitative nature of g(r) as a function of r for a low density? How does g(r) change as the density is increased? How does g(r) for a system of hard disks compare to g(r) for a system of Lennard-Jones particles at the same density?
</li>

</ol> 

<p class="header_title">Java Classes</p>

<ul>

<li>HD</li>
<li>HDApp</li>

</ul>

<p class = "small">Updated 9 May 2008.</p>

</body>
</html>